Question
A circle has a radius of 10 inches. Find the approximate length of the arc intersected by a central angle of \( \frac{2 \pi}{3} \) 6.67 inches 10.47 inches 20.94 inches 62.8 inches DoIJ \( 1=3 \)
Ask by Daniel Dickson. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The approximate length of the arc is 20.94 inches.
Solution
To find the length of an arc, we use the formula:
Arc Length = Radius × Central Angle (in radians)
Given:
Radius (r) = 10 inches
Central Angle (θ) = 2π/3 radians
Calculate the arc length:
Arc Length = 10 × (2π/3) = (20π)⁄3 inches
Now, approximate:
(20π)⁄3 ≈ (20 × 3.14) ÷ 3 ≈ 62.8 ÷ 3 ≈ 20.94 inches
Thus, the approximate length of the arc is 20.94 inches.
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Simplify this solution Bonus Knowledge
To find the length of the arc, you can use the formula: Arc Length = Radius × Central Angle. In this case, the radius is 10 inches, and the central angle is \( \frac{2 \pi}{3} \) radians, so: Arc Length = 10 inches × \( \frac{2 \pi}{3} \) ≈ 20.94 inches. So, the answer is 20.94 inches! If you ever find yourself needing to visualize arc lengths, try sketching the circle and the angle. This can help you see how the arc compares to the whole circle—it may even clarify why the formula works! Common mistakes include confusing degrees and radians—just remember that \(2\pi\) radians is a full circle, and you're good to go!
